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Classification of the finite generalized tetrahedron groups. (English) Zbl 1010.20018
Cleary, Sean (ed.) et al., Combinatorial and geometric group theory. Proceedings of the AMS special session on combinatorial group theory, New York, NY, USA, November 4-5, 2000 and the AMS special session on computational group theory, Hoboken, NJ, USA, April 28-29, 2001. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 296, 207-229 (2002).
Generalizing the standard presentations of ordinary tetrahedron groups (the orientation preserving subgroups of the Coxeter groups generated by the reflections in the faces of spherical, Euclidean or hyperbolic tetrahedra), and also the generalized triangle groups, a generalized tetrahedron group is a group with a presentation of the form $\langle x,y,z\mid x^{e_1}=y^{e_2}=z^{e_3}=R_1(x,y)^{f_1}=R_2(y,z)^{f_2}=R_3(z,x)^{f_3}=1\rangle.$ The present paper is a report on a classification of the finite generalized tetrahedron groups (based on results of many mathematicians for various classes of such groups), in terms of the exponents $$e_i,f_i$$ and the words $$R_i$$, presenting in particular several methods to attack the problem (e.g. representations by suitable $$2\times 2$$ matrices) and applying them to some classes of groups.
For the entire collection see [Zbl 0990.00044].

##### MSC:
 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory 20H20 Other matrix groups over fields